Let $P(x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?

The statement in option (A) represents the standard Principle of Mathematical Induction over the set of natural numbers $\mathbb{N} = \{0, 1, 2, \dots\}$:

• The base case establishes $P(0)$.

• The inductive step guarantees that if $P(x)$ holds for any natural number $x$, then $P(x+1)$ also holds.

• By induction, $P(x)$ must be true for all natural numbers $x$.

Why the other options are incorrect:

• (B) uses $P(x) \implies P(x-1)$, which goes backwards and cannot prove statements for numbers greater than $0$.

• (C) and (D) use $P(1000)$ as the base step, which only establishes the property for numbers $\le 1000$ or $\ge 1000$ respectively, failing to cover all natural numbers $\forall x P(x)$.

Therefore, (A) is the only universally valid statement.